Demographic Models

Exponential growth of a population is the rate of population growth in situations where food and resources are unlimited. Exponential growth assumes ideal conditions, which are impossible in nature; conditions might be ideal for a period of time (and exponential growth can occur), but they will not be ideal indefinitely. For example, a female house mouse (mouse A) and her mate establish a home in an old granary. Food is consistently available, conditions are excellent for reproduction, and there is no risk of predation. Mouse A produces a litter of four pups, two females and two males. Ten weeks later, mouse A and her two daughters (B and C) all deliver litters of healthy pups. Of the 15 pups, 8 are female. In an additional 10-week period, all females—through mouse K—produce new litters. Now, the mouse population has exploded from 2 to 77 mice in five months.

Scientists can create ideal conditions by controlling the breeding of individuals within a controlled population, ensuring adequate food, providing medical attention to deal with disease, and preventing predation. This is the method by which zoos around the world monitor controlled breeding of endangered species. The American bison is a case study in exponential growth patterns. At one point, the bison population on the Great Plains ranged between 15 million and 100 million individuals, but human interference reduced this massive population to about 1,000 bison by 1888. Conservation efforts began in the early 1900s. Through assisted breeding and protection, the bison population has expanded exponentially to nearly 450,000 individuals in 2010. The calculation for an exponential growth model uses the equation $\frac{\Delta N}{\Delta t}=rN$ where ∆N is the change in the population divided by ∆t, the change in time, and rN is the rate of increase. This equals the rate of increase in a population. In the case of the mice, the change in number (∆N) would be 75 mice (77 offspring minus 2 original parents) divided by the change in time (∆t), 20 weeks, which equals a population rate increase of 3.75 mice per week. With the American bison, the 1888 population of 1,300 increased to 450,000 over a period of 122 years, or 448,700 bison divided by 122 years, which equals a rate of 3,678 bison per year.

A logistic growth model indicates how a population grows more slowly when it reaches the carrying capacity of its environment. The carrying capacity of an ecosystem varies as the food supply varies. In the Arctic, snowy owls, Arctic foxes, and stoats (a type of weasel) feed on lemmings. When lemming populations rise, so do the populations of owls, foxes, and stoats. When food is scarce, lemming populations decline, and predator populations decline accordingly because their food source is less available. In temperate forests, wildfires can destroy grasses, forbs, vines, and wildflowers and reduce tree populations. Ash from the fires nourishes the soil, and exponential growth follows the disruption of the original growth pattern.

Human populations have also encountered limiting factors to population growth. Specifically, epidemic diseases have dramatically changed worldwide populations in the past. The Black Plague wiped out 30–50% of the human population between 1347 and 1351. World War I caused 18 million deaths and was quickly followed by the Spanish flu pandemic, which killed 20–50 million people.

Adjustments to the exponential growth equation provide a mathematical solution to a logistic equation. Begin with the equation $\frac{\Delta N}{\Delta t}=rN$ , where $\frac{\Delta N}{\Delta t}$ is the rate of change in the population. The term $\left(\frac{(K-N)}K\right)$ represents the carrying capacity of the ecosystem, and the term rN represents the population size. The equation for logistic growth is $\frac{\Delta N}{\Delta t}=rN\;\left(\frac{(K-N)}K\right)$ . The equation adjusts for the limited resources that may affect reproduction or growth rates. As an example, consider a population of seals. A small group of seals colonize a new island. The seal population grows exponentially, following the exponential growth equation. Over time, the number of seals nears carrying capacity of the island: seals must compete with each other for limited food and places to raise their young, and not all of them survive this competition. The population growth plateaus, following the logistic growth equation.